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15 include "basic_2/notation/relations/predsnstaralt_5.ma".
16 include "basic_2/computation/cpxs_cpxs.ma".
17 include "basic_2/computation/lpxs.ma".
19 (* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
21 (* alternative definition *)
22 definition lpxsa: ∀h. sd h → relation3 genv lenv lenv ≝
23 λh,g,G. lpx_sn … (cpxs h g G).
25 interpretation "extended parallel computation (local environment, sn variant) alternative"
26 'PRedSnStarAlt h g G L1 L2 = (lpxsa h g G L1 L2).
28 (* Main properties on the alternative definition ****************************)
30 theorem lpxsa_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡➡*[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
31 /2 width=1 by lpx_sn_LTC_TC_lpx_sn/ qed-.
33 (* Main inversion lemmas on the alternative definition **********************)
35 theorem lpxs_inv_lpxsa: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1⦄ ⊢ ➡➡*[h, g] L2.
36 /3 width=3 by TC_lpx_sn_inv_lpx_sn_LTC, lpx_cpxs_trans/ qed-.
38 (* Alternative eliminators **************************************************)
40 lemma lpxs_ind_alt: ∀h,g,G. ∀R:relation lenv.
43 ⦃G, K1⦄ ⊢ ➡*[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 →
44 R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
46 ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L1 L2.
47 /3 width=4 by TC_lpx_sn_ind, lpx_cpxs_trans/ qed-.